Alignment has greatly improved large language models (LLMs)' output quality at the cost of diversity, yielding highly similar outputs across generations, especially in open-ended generation tasks. We propose Base-Aligned Model Collaboration (BACo), an inference-time token-level model collaboration framework that dynamically combines a base LLM with its aligned counterpart to optimize diversity and quality. Using uncertainty and content-based signals, BACo employs routing strategies to determine, at each token, which model to decode from. Prior diversity-promoting methods often improve diversity at the expense of quality or require expensive decoding or post-training. In contrast, BACo achieves both high diversity and quality post hoc within a single pass, while offering strong controllability. We introduce a family of effective routing strategies and evaluate them across three open-ended generation tasks with 13 diversity and quality metrics. BACo consistently surpasses state-of-the-art inference-time baselines. With our best router, BACo achieves a 21.3% joint improvement in diversity and quality, which is further supported by human evaluations. Overall, our results demonstrate that collaboration between base and aligned models provides an effective and controllable mechanism for optimizing the diversity-quality trade-off.